Professur Stochastik
Research Topics
Spectral Theory
of selfadjoint operators, second order elliptic differential operators on Euclidean spaces and manifolds, Laplace(-Beltrami) operators, Schrödinger operators, discrete Schrödinger operators and more general finite difference operators on lattices and general groups and graphs
Schrödinger operators
selfadjointness, spectral theory, behaviour of eigensolutions, low-lying eigenvalues, Allegretto-Piepenbrink theory for general operators
Random operators modelling disordered systems
e.g. random Schrödinger operators, Anderson models, random Hamiltonians with correlations, quantum percolation models, general percolation Hamiltonians
Mathematical theory of Anderson localisation
proof of localisations via multiscale analysis and the fractional moment method, Wegner estimates, Lifschitz tails and initial scale estimates, decoupling properties of random operators, non-monotone parameter dependence of operators
Ergodic Theory
in particular as a tool to study the integrated density of states, Banach-space valued ergodic theorems, uniform approximation via ergodic theory
Percolation theory
percolation on Cayley graphs, sharpness of the phase transition, distribution of the cluster size, Laplacians on percolation clusters
Asymptotic aspects of geometric group theory
random walks on Cayley graphs, quasi-isometries, Laplacians on graphs, their spectral distribution function, eigenfunctions
See also the web page of the Emmy-Noether-Project.